CERI 7315/8315 Computational Methods for Geodynamics

Instructor: Eunseo Choi

Time and Location: MWF, 10:20 - 11:15 am.

Contact: echoi2@memphis.edu

Grades

• Topical homeworks: 50 %
• Term project: 30 %
• Quizzes: 20 %

Office Hours

• To be arranged individually
• Chatting and meeting via MS Teams preferred

Goal and Objectives

This course aims to enable students to understand the basics of the finite element method (FEM), a versatile numerical method for solving partial differential equations.

After taking this course, students will be able to use or modify as necessary the existing community finite element codes (e.g., CIG codes) for their geophysical research.

To achieve the goal, we will

1. review the fundamental governing equations in continuum mechanics,

2. have under-the-hood understanding of finite element method,

3. gain hands-on experience with common procedure and useful practices in computational research,

4. use one of the open-source FEM codes, possibly after modifications, for their term project.

References and Online Resources

No required textbook but parts of the references listed below will be used.\

Reference texts¹

• Continuum mechanics:

  • $^{\dagger}$ Tadmor, E. B., Elliott, R. S., and Miller, R. E. (2012). Continuum Mechanics and Thermodynamics: From Fundamental Concepts to Governing Equations. Cambridge University Press, Cambridge
  • Holzapfel, G. A. (2000). Nonlinear solid mechanics : a continuum approach for engineering. Wiley, Chichester ; New York
  • Malvern, L. E. (1977). Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Upper Saddle River, New Jersey

• Fundamental numerical techniques

  • $^{\dagger}$ Quarteroni, A., Sacco, R., and Saleri, F. (2000). Numerical Mathematics. Springer-Verlag, New York
  • $^{\dagger}$ Zienkiewicz, O. C., Zhu, J. Z., and Taylor, R. L. (2013). The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, 7th edition
  • $^{\dagger}$ Ismail-Zadeh, A. and Tackley, P. (2010). Computational Methods for Geodynamics. Cambridge University Press
  • $^{\dagger}$ Gerya, T. (2009). Introduction to Numerical Geodynamic Modelling. Cambridge University Press, New York

• Geodynamics:

  • Turcotte, D. L. and Schubert, G. (2002). Geodynamics. Cambridge University Press, New York, 2nd edition
  • $^{\dagger}$ Schubert, G., Turcotte, D. L., and Olson, P. (2001). Mantle Convection in the Earth and Planets. Cambridge University Press, Cambridge
  • Davies, G. F. (1999). Dynamic Earth: Plates, Plumes, and Mantle Convection. Cambridge University Press, Cambridge

Online resources

• The web sites listed below will help you get familiar with the command line-based work environment and other useful tools for computational research.

  • How to work on a Linux(-like) system especially when you are new to it:
    https://developer.ibm.com/technologies/linux/tutorials/;
    search for tutorials with keywords "LPIC-1" and "exam 1"

  • Lessons on BASH, Python and Git by Software Carpentry:
    https://software-carpentry.org/lessons/

  • Programming languages (C, Python, etc) and parallel computing (OpenMP, MPI, GPU etc)
    https://cvw.cac.cornell.edu/topics

Term projects

• Students carry out a reasonably small but non-trivial project relevant to the course's goal and objectives.

• They should use GitHub to manage their projects and products as sharable and reusable resources.

• A project topic will be decided individually based on students' interests and needs.

• Possible topics:

  • Consider in a global-scale mantle convection model the effects of centrifugal acceleration in addition to the typical geocentric gravity

  • Reproduce and possibly improve a published work on computational methods.

  • Parallelize an existing code with a directive-based approach such as OpenMP and OpenACC and assess the performance improvement

  • Introduce recent advances in physics-informed neural networks (PINNs)

Course Outline

• Week 1: A short review of continuum mechanics

• Week 2: Numerical toolbox - Principles of numerical mathematics

• Week 3: Numerical toolbox - Interpolation: Lagrange polynomial

• Week 4: Numerical toolbox - Interpolation: Piecewise Lagrange polynomal interpolation in 2D

• Week 5: Numerical toolbox - Solving linear equations: Basic stability analysis and direct method

• Week 6: Numerical toolbox - Solving linear equations: Iterative methods and conjugate gradient method

• Week 7: Numerical toolbox - Solving linear equations: Krylov subspace methods. Solving non-linear systems

• Week 8: Numerical toolbox - Approximating function derivatives: Finite difference and interpolation-based approach

• Week 9: Numerical toolbox - Approximating function derivatives: Orthogonal polynomials and weight functions

• Week 10: Numerical toolbox - Numerical integration: Gauss and Gauss-Lobatto quadrature formula

• Week 11: Basic finite element method - Examples of PDEs, Weak forms and variational principles

• Week 12: Basic finite element method - Walkthrough with the Poisson eq. in 1D

• Week 13: Basic finite element method - Extension to 2D and 3D

• Week 14: Basic finite element method - Solving time-dependent PDEs

• Week 15: Selected Topics

  • Elastic deformation: Static and Dynamic
  • Basic parallel computing
  • Introduction to open-source codes: PyLith, ASPECT, FEniCS, or DES3D

Footnotes

1. $^{\dagger}$ means that the UofM Library has an ebook version. ↩